Ordinary Annuity

An ordinary annuity is a series of equal payments made or received at the end of each period, such as each month or year, for a fixed number of periods. Examples include most loan and mortgage repayments, bond interest coupons, and lease payments made in arrears. It is the standard form of annuity assumed in financial calculators and spreadsheets unless the user specifies otherwise. It is also known as an annuity in arrears.

The ordinary annuity is contrasted with an annuity due, in which each payment is made at the beginning of the period, such as rent paid in advance. Because each payment in an annuity due arrives one period earlier, it is worth more in present value terms. Both are central tools in the time value of money, which holds that a dollar received today is worth more than a dollar received later.

The present value, what a series of future payments is worth today, is:

• $PV$ is the present value of the annuity.
• $PMT$ is the payment made at the end of each period.
• $r$ is the interest or discount rate per period.
• $n$ is the number of periods.

Suppose a business will receive $10,000 at the end of each year for five years, and the appropriate rate is 7 percent. The annuity factor is (1 minus 1.07 to the power of negative 5) divided by 0.07, which is about 4.1002. The present value is $10,000 times 4.1002, or about $41,002. If the same payments were received at the start of each year, as an annuity due, the value would be $41,002 times 1.07, or about $43,872.

The future value, what the payments will grow to by the end of the last period, is:

Using the same figures, the future value factor is (1.07 to the power of 5, minus 1) divided by 0.07, which is about 5.7507, so the payments grow to about $57,507. The first payment earns interest for four years and the last earns none, because it arrives on the final day. This calculation is covered in more detail in the entry on the future value of an ordinary annuity.

• Loans: the level payment on an installment loan is found by rearranging the present value formula, and each payment is then split between interest and principal.
• Bonds: the price of a bond is the present value of its coupon payments, an ordinary annuity, plus the present value of the face amount at maturity, discounted at the market rate.
• Leases: lessees measure lease liabilities as the present value of future payments; payments in arrears are an ordinary annuity.
• Capital budgeting: projects with equal annual cash flows are evaluated with annuity factors in net present value (NPV) analysis.
• Sinking funds and savings: the future value formula shows how much regular deposits will accumulate to, for example in a bond sinking fund.

The most frequent error is mixing up timing: using the ordinary annuity formula for payments made in advance understates their value. Another is mismatching the rate and the period: monthly payments require a monthly rate, so an annual rate of 6 percent becomes 0.5 percent per month, and a five-year loan has 60 periods. A third is applying annuity formulas to payments that are not equal, such as payments that grow each year, which require a growing annuity formula or a period-by-period calculation. Finally, the rate used should reflect the risk of the cash flows; using a low rate for uncertain payments overstates their value. Financial calculators handle these differences through settings for payment timing, often labeled "END" for an ordinary annuity and "BEGIN" for an annuity due, and the same concepts underlie compound interest calculations generally.